Raj wants to travel from point P to point R. Using the available routes and vehicles, what is the minimum time required for Raj to cover this distance?

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**Solution:**

1. To find the minimum time, we first identify the shortest possible route from P to R.

2. The two possible routes are:
   - Route (i): P -> Q -> S -> R
   - Route (ii): P -> S -> R

3. From the given data, the direct distance from P to S is 520 km, and from S to R is 600 km.

4. The speeds of the vehicles on these routes are:
   - Taxi from P to S: 80 km/h
   - Bike from S to R: 25 km/h

5. Calculate the time taken for each segment on the shortest route (P -> S -> R):
   - Time from P to S = Distance / Speed = $\frac{520}{80} = 6.5$ hours
   - Time from S to R = $\frac{600}{25} = 24$ hours

6. Total time for the route P -> S -> R = $6.5 + 24 = 30.5$ hours

7. Since this is the minimum time compared to other possible routes, the answer is 30.5 hours.

**Answer:** 30.5 hours

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